I've wondered about the following question :
Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such that $x_n$ converges to $x$ with respect to $\|\cdot\|_1$, $x_n$ converges to $y$ with respect to $\|\cdot\|_2$, but $x \neq y$?
Obviously, this would imply that $\|\cdot\|_1$ and $\|\cdot\|_2$ are not equivalent. In fact, these two statements are equivalent, which is a consequence of the Open Mapping Theorem.